Axiom-based Probabilistic Description Logic

Martin Unold

and Christophe Cruz

Laboratoire d’Informatique de Bourgogne, EA 7534, 9 Avenue Alain Savary, BP 47870, 21078 Dijon Cedex, France

Keywords:

Probabilistic Description Logic, Uncertainty, Knowledge Base.

Abstract:

The paper proposes a new type of probabilistic description logics (p-DLs) with a different interpretation of

uncertain knowledge. In both approaches (classical state of the art approaches and the approach of this paper),

probability values are assigned to axioms in a knowledge base. While In classical p-DLs, the probability value

of an axiom is interpreted as the probability of the axiom to be true in contrast to be false or unknown, the

probability value in this approach is interpreted as the probability of an the axiom to be true in contrast to

other axioms being true. The paper presents the theory of that novel approach and a method for the treatment

of such data. The proposed description logic is evaluated with some sample knowledge bases and the results

are discussed.

1 INTRODUCTION

The management of uncertainty in description logics

has received a lot of attention, due to the development

of the semantic web. There should be a level of trust

for every piece of information found online and when

combining this information to gain further implicitly

stored knowledge, the involved uncertainty should be

taken into account. Therefore probabilistic descrip-

tion logics have been developed in a couple of vari-

ants. Usually they see each probabilistic axiom within

the same knowledge base separately from all others,

i.e. the probability of that axiom should be valid in

any scenario.

Nevertheless, in some cases, the probability is

only estimated or the source of uncertainty allows

only to state a general value for the uncertainty of all

the information gained from that source. For exam-

ple, the two statements

• A is true with a probability of 90%

• A is false with a probability of 10%

work perfectly together and would result in a consis-

tent knowledge base. If the statements are changed

• A is true with a probability of 90%

• A is false with a probability of 50%

https://orcid.org/0000-0003-2913-2421

https://orcid.org/0000-0002-5611-9479

they do not hold in a probabilistic knowledge base.

Only some possibilistic approaches would be able to

handle them. However, the two statements

• A is true with a probability of 90%

• A is false with a probability of 90%

would not work together in any case. Even though

this looks like a huge contradiction, such cases might

appear in several scenarios. For instance, when merg-

ing two crisp knowledge bases K

and K

, K

might

contain one of these statements and K

the other one.

For sure, they are inconsistent together and therefore

at least one of the knowledge bases must contain false

information. If it is known, that approximately 90%

of the knowledge in K

and K

is correct, respec-

tively, one might not have any other choice than as-

signing 90% to each axiom of both knowledge bases

to express the probability. The trust of 90% could be a

result from earlier observations about the knowledge

from the same source.

Generally, it might be a more realistic assump-

tion that there exists only one correct world and not

several possible worlds, each with a certain probabil-

ity or possibility. Indeed, it is the knowledge about

the correct world, which might be wrong sometimes,

rather than the world itself being true only with a cer-

tain probability or possibility. For instance, a pater-

nity testing has an accuracy of 99.99%. This value

is determined by observations when applying the test

to people with known paternity. Hence, a statement

like ”John is the father of Mary with a probability of

99,99%” does not state, that in 9999 of 10000 a paral-

Unold, M. and Cruz, C.

Axiom-based Probabilistic Description Logic.

DOI: 10.5220/0008351103830389

In Proceedings of the 11th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management (IC3K 2019), pages 383-389

ISBN: 978-989-758-382-7

383

lel universes the statement is true and in 1 it is false. It

rather states, that 9999 of 10000 comparable parent-

ing testings are correct and 1 is incorrect. But there is

only one world.

In such cases, the probability does not state the

probability of a statement to be true in contrast to be-

ing false or unknown. It rather states the probabil-

ity of a statement to be true in contrast to the other

statements in the same environment or from the same

source. This type of interpretation is neither handled

by probabilistic nor by possibilistic description log-

ics. This paper presents a different way of dealing

with knowledge in such situations. It starts with a

related work and a preliminaries section, where ba-

sically other ways of creating probabilistic or possi-

bilistic knowledge bases are presented, respectively.

The novel approach is then explained in section 4 and

section 5 shows in which cases it is more useful. Fi-

nally, a conclusion summarizes the paper and gives an

outlook to further developments.

2 RELATED WORK

A description logic is used as a formalism to represent

knowledge about a special domain. It easily allows to

link other information to knowledge in the semantic

web, as explained in (Baader et al., 2005). Reason-

ing programs allow to infer implicitly stored infor-

mation from explicit knowledge and inference rules.

These programs can also decide if a speciﬁc part of

the knowledge is logically consistent or not. Common

decision problems also include those, that are typi-

cally used for queries on relational databases, such as

instance or relation checking.

There exists a huge variety of description logics

with different expressiveness. An overview about this

variety is shown in (Zolin, 2013). The expressiveness

is determined by the amount of symbols allowed in

the syntax of a particular description logic. In this sec-

tion, the most common and the ones that are relevant

in the following chapters are explained. A more de-

tailed introduction to description logics can be found

in (Baader et al., 2003).

The extension of a description logic by the man-

agement of uncertainty results in a probabilistic de-

scription logic. In contrast to a classical one, which

is also called crisp. Regardless of the expressiveness,

each type of description logic could be extended to a

probabilistic one. When modeling a probabilistic de-

scription logic, the knowledge about the validity of

the axioms is limited, i.e. it is unknown if an axiom

is true or false, yet the axioms themselves are clearly

true or false and nothing in between. An introduction

to probabilistic logic can be found in (Nilsson, 1986).

Probability and fuzziness (also called vagueness)

seem very similar on ﬁrst sight, since both contain

values in the range between 0 and 1 attached to crisp

axioms in the knowledge base of a description logic.

Nevertheless, they both work completely different

and a detailed clariﬁcation about the differences is

done in (Dubois and Prade, 2001). An introduction to

both types of extensions of description logics is done

in (Lukasiewicz and Straccia, 2008). The structure

of a probabilistic description logic is rather the same

as in the crisp case and the probabilistic interpreta-

tion maps all possible crisp interpretations to a certain

probability.

A major problem with probabilistic description

logics is the complexity (Lukasiewicz, 2008), in gen-

eral cases computations require exponential time, be-

cause many possible worlds need to be taken into ac-

count. Hence, a lot of research focused on the re-

duction of complexity, such as (Klinov, 2008), or to

ﬁnd more efﬁcient solutions for more speciﬁc cases,

such as (Riguzzi et al., 2015). The latter approach as-

sumes that all axioms within a knowledge base are in-

dependent of each other, while in this paper all axioms

within a knowledge base are assumed to be highly de-

pendent. In (Niepert et al., 2011) a log-linear model

is used to do reasoning on uncertain knowledge.

One of the ﬁrst attempts to add the management

of uncertainty to description logics is done by (Koller

et al., 1997), which was developed further by (Giugno

and Lukasiewicz, 2002), where the complexity of rea-

soning algorithms has been analyzed. The develop-

ment of reasoning programs has been subject of sev-

eral studies (Lukasiewicz, 2007) and also the devel-

opment of standards for the semantic web is of im-

portance for applications. The management of un-

certainty could be integrated to the semantic web by

extensions of OWL or RDF. Examples are found in

(Ding et al., 2006). Unfortunately, there is no W3C-

standard available till now (Carvalho et al., 2017).

Anyway, the solution for dealing with probabilities

in description logics presented in this paper should

be treated carefully when combining with other data,

that use a different understanding of uncertainty.

3 PRELIMINARIES

Each description logic allows the existence of knowl-

edge bases, where all the information about a speciﬁc

domain is stored. Typically, the knowledge is divided

into two parts, which are called the terminological ax-

ioms (TBox) and the assertional axioms (ABox). The

TBox consists of general rules, such as ”Each City

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384

has a Location”. The ABox consists of speciﬁc infor-

mation for single instances, such as ”Paris is a City”.

The reasoner should then be able to infer the implicit

information, that ”Paris has a Location”.

A knowledge base K is a collection of axioms.

Usually, they are divided into two parts, so that a

knowledge base becomes a tuple K = (T , A). It con-

sists of a TBox T and an ABox A. A TBox T is a set

of general concept inclusions and an ABox A is a set

of assertional axioms. A general concept inclusion is

an expression of the format C v D, where C,D ∈ C

are concepts. An assertional axiom is either an ex-

pression of the format a : C (concept assertion) or of

the format (a,b) : R (role assertion), where a,b ∈ N

are named individuals, C ∈ C is a concept and R ∈ N

is a role.

In this context, an ordered triple of arbitrary pair-

wise disjoint sets (N

) is called a signature.

The sets contain named concepts, named roles and

named individuals, respectively. The set of concepts

C over a signature (N

) is the smallest possi-

ble set, that fulﬁlls some of the following conditions.

C ∈ N

=⇒ C ∈ C

> ∈ C

⊥ ∈ C

∈ C =⇒ C

∈ C

∈ C =⇒ C

∈ C

C ∈ C =⇒ ¬C ∈ C

C ∈ C,R ∈ N

=⇒ ∀R.C ∈ C

C ∈ C,R ∈ N

=⇒ ∃R.C ∈ C

Each element C ∈ C is called a concept and it de-

pends on the expressiveness of the description logic, if

even more conditions must be fulﬁlled, i.e. more sym-

bols are allowed. The description logic ALC allows

only atomic roles and the introduced concepts. The

basic version of ALC neither allows role inclusion

axioms nor general concept inclusions, i.e. only as-

sertional axioms are allowed. Description logics with

more expressiveness could allow further roles, con-

cepts and even an entirely different additional formal-

ism.

So far, it is only a collection of symbols. The in-

teresting part is the semantics of these axioms. The

meaning of each axiom is expressed by stating the

membership to each concept and role. These state-

ments are done by an interpretation, which maps the

signature to some elements of the universe of dis-

course, which is the domain of the knowledge, that

is modeled in a particular knowledge base.

Formally, an interpretation I = (∆

,·

) over a

signature (N

) consists of a set ∆

, which is

called domain or universe of discourse, and a func-

tion ·

, that maps ...

• ... each named individual a ∈ N

to an element of

the domain a

∈ ∆

• ... each concept C ∈ C to a subset of the domain

⊆ ∆

• ... each role R ∈ R to a set of tuples of domain

elements R

∈ ∆

× ∆

The interpretation function ·

must also fulﬁll the

following conditions.

δ ∈ >

⇐⇒ δ ∈ ∆

δ ∈ ⊥

⇐⇒ δ /∈ ∆

δ ∈ (C

)

⇐⇒ δ ∈ C

∨ δ ∈ C

δ ∈ (C

)

⇐⇒ δ ∈ C

∧ δ ∈ C

δ ∈ (¬C)

⇐⇒ δ /∈ C

δ ∈ (∀R.C)

⇐⇒

∀δ

∈ ∆

,(δ,δ

) ∈ R

: δ

∈ C

δ ∈ (∃R.C)

⇐⇒

∃δ

∈ ∆

,(δ,δ

) ∈ R

: δ

∈ C

Of course, the conditions depend on the existence

of the corresponding type of concepts and roles, i.e.

only those conditions must be fulﬁlled, where the cor-

responding type of concepts and roles are allowed by

the description logic.

A knowledge base K is free of contradictions, if

there exists an interpretation I , which is a model for

all axioms of the knowledge base. An interpretation

I is a model of a general concept inclusion C v D

(denoted: I |= (C v D)) if and only if C

⊆ D

. An

interpretation I is a model of a role assertion (a,b) : R

(denoted: I |= ((a, b) : R)) if and only if (a

) ∈ R

An interpretation I is a model of a concept assertion

a : C (denoted: I |= (a : C)) if and only if a

∈ C

An interpretation I is a model of a TBox T (denoted:

I |= T ) if and only if I |= φ for all general concept

inclusions φ ∈ T . An interpretation is a model of an

ABox A (denoted: I |= A) if and only if I |= φ for

all assertional axioms φ ∈ A. An interpretation I is a

model of a knowledge base K , if I is a model of all

its boxes. Sometimes the phrase ”in I holds” is used

instead of ”I is a model of”.

For a probabilistic description logic, the set of

possible worlds

W =

{

I | I is an interpretation over (N

)

}

consists of all interpretations over a given signature

). An element of this set I ∈ W is called a

possible world. A mapping π : W → [0, 1] with

∑

I ∈W

π(I ) = 1

Axiom-based Probabilistic Description Logic

385

is called a probability distribution of possible worlds

over (N

). It has to be noted, that the deﬁni-

tion of W and π is not dependent on a given interpre-

tation or knowledge base, only an elementary descrip-

tion is necessary. This is important, especially when

it comes to the development of algorithms, since there

are a lot of possible worlds on ﬁrst attempt.

An uncertain knowledge base will be intro-

duced later to restrict π. Sometimes π is not de-

ﬁned as a probability distribution and the constraint

∑

I ∈W

π(I ) = 1 is replaced by max

I ∈W

π(I ) = 1. With

that constraint, one gets a possibility value for each

world to be the true one.

A probabilistic knowledge base K is a collection

of axioms. They could be probabilistic, but also crisp

axioms, i.e. with a probability of 1. As in the crisp

case, the knowledge base is divided into two parts

K = (T ,A). It consists of a TBox T and an ABox A.

A TBox T is a set of (probabilistic) general concept

inclusions and an ABox A is a set of (probabilistic)

assertional axioms. A probabilistic general concept

inclusion is an expression of the format p :: C v D,

where C,D ∈ C are concepts and p ∈ [0,1] is the

probability of the axiom to be true. A probabilistic

assertional axiom is either an expression of the for-

mat p :: (a : C) (concept assertion) or of the format

p :: ((a,b) : R) (role assertion), where a,b ∈ N

are

named individuals, C ∈ C is a concept, R ∈ N

is a

role and p ∈ [0,1] is the probability of the axiom to be

true. For p = 1, an axiom has exactly the crisp mean-

ing. An axiom with p = 0 is obsolete because it does

not contain any information.

The set of concepts C is deﬁned in exactly the

same way as in the crisp case. Also the interpreta-

tion I is deﬁned in exactly the same way. The only

difference is, that it is called a possible world in the

context of probabilistic description logics.

A probabilistic knowledge base K is free of con-

tradictions, if there exists a probability distribution

of possible worlds π, which is a model for all ax-

ioms of the probabilistic knowledge base. A proba-

bility distribution of possible worlds π is a model of

a model of a probabilistic general concept inclusion

p :: C v D (denoted: π |= p :: C v D) if and only if

∑

I |=CvD

π(I ) = p. It is a model of a probabilistic role

assertion p :: (a,b) : R (denoted: π |= p :: (a, b) : R) if

and only if

∑

I |=(a,b):R

π(I ) = p. And it is a model

of a probabilistic concept assertion p :: a : C (de-

noted: I |= p :: a : C) if and only if

∑

I |=a:C

π(I ) = p.

A probability distribution of possible worlds π is a

model of a TBox T (denoted: π |= T ) if and only if

π |= p :: φ for all (probabilistic) general concept in-

clusions p :: φ ∈ T . And it is a model of an ABox

A (denoted: π |= A) if and only if π |= p :: φ for all

(probabilistic) assertional axioms p :: φ ∈ A. A prob-

ability distribution of possible worlds π is a model of

a probabilistic knowledge base K , if it is a model of

all its boxes.

It has to be noted, that there are also other options

to deﬁne a probabilistic description logic. There is

also the similar possibilistic logic, which allows to

state possibilities and necessities for each axiom and

the probability distribution is replaced by the maxi-

mum possibility for each world. These options are

not part of further discussion at this point, since the

proposal of this paper is another option, anyway.

4 PROPOSAL

In contrast to the probabilistic description logics in-

troduced in the previous sections, the probability at-

tached to an axiom p :: φ is interpreted differently in

this proposal. Here, p is the probability of φ being

true not in contrast to being false or unknown other-

wise, but to be true as compared to other axioms of

the same knowledge base.

An axiom-based probabilistic knowledge base is a

tuple K = (T ,A), that consists of probabilistic gen-

eral concept inclusions (p :: C v D) ∈ T , probabilistic

concept assertions (p :: a : C) ∈ A and probabilistic

role assertions (p :: (a, b) : R) ∈ A. To ease the no-

tation, φ is denoted for any type of crisp axiom and

(p :: φ) ∈ K is used to state that an axiom is part of a

knowledge base, although the axioms are members of

T or A.

The basic idea of this proposal is, that within an

axiom-based probabilistic knowledge base, the ratio

between correct and incorrect axioms should be ac-

cording to their probability, respectively. Therefore,

the sum of all remaining probabilities of all correct

statements should be equal (or greater than) the sum

of all probabilities of all incorrect statements, which

is measured as conﬁdence. A conﬁdence function

σ : W → R assigns to every possible world I ∈ W a

value that states the conﬁdence of the world to be the

true one.

σ(I ) =

|K |







∑

p::φ∈K

I |=φ

(1 − p) −

∑

p::φ∈K

I 6|=φ







(1)

The value for conﬁdence of an interpretation

ranges from −1, which indicates a totally wrong in-

terpretation, to 1. A value of around 0 or maybe a bit

higher indicates an interpretation, that is very likely

to be the true one. If the value is very high, i.e. close

KEOD 2019 - 11th International Conference on Knowledge Engineering and Ontology Development

386

to 1, this is rather an indicator of a not well designed

knowledge base, that is not restrictive enough.

As an example, consider the knowledge base K =

(T ,A) with the following axioms.

T = {0.75 :: C v D}

A = {0.75 :: a : C,

0.75 :: a : C tD,

0.75 :: a : ¬D}

The knowledge base consists of 4 axioms. To each

a probability of 0.75 is assigned, which means that

in average one of these axioms is false in the way of

interpreting probability within this paper. It has to

be noted, that each probability could be any value be-

tween 0 and 1. For the example knowledge base, there

exist four different interpretations.

: a

∈ C

∈ D

: a

∈ C

/∈ D

: a

/∈ C

∈ D

: a

/∈ C

/∈ D

The conﬁdence of each interpretation in W =

} can be calculated in the following way.

σ(I

) =

(0.25 + 0.25 + 0.25 −0.75) = 0

σ(I

) =

(−0.75 +0.25 + 0.25 +0.25) = 0

σ(I

) =

(0.25 − 0.75 + 0.25 −0.75) = −0.25

σ(I

) =

(0.25 − 0.75 − 0.75 +0.25) = −0.25

This solution shows that either I

or I

are very likely

to be the correct worlds. It also gives a hint to which

axioms might be the wrong ones in the knowledge

base (here it is C v D or a : ¬D). In a classical proba-

bilistic description logic, this example would be con-

sidered inconsistent. Also, a possibilistic logic would

either suffer from the drowning problem and remove

all axioms from the knowledge base or if other strate-

gies are used still not ﬁnd a totally possible world.

There are several algorithms to handle probabilis-

tic logics. The consideration of all possible worlds

is very time consuming, since the amount of possible

worlds increases exponentially in the amount of ax-

ioms in the knowledge base. Therefore some more ef-

ﬁcient algorithms have been designed, they have dis-

advantages sometimes and might not work for an ar-

bitrary knowledge base. For the proposed description

logic, an optimization problem has to be solved, since

the goal is to ﬁnd the possible world with the highest

conﬁdence. Therefore the development of suitable al-

gorithms, such as evolutionary algorithms, is subject

to future work. Nevertheless, it might be easier to ﬁnd

an efﬁcient solution for an optimization problem as

compared to solve a huge linear system of equations,

which is required for (classical) probabilistic descrip-

tion logics.

If all possible worlds have a negative conﬁ-

dence, the knowledge base might be inconsistent, i.e.

the information within the axiom-based probabilistic

knowledge base is wrong with respect to the probabil-

ity values. A small negative value for the highest con-

ﬁdence might still be called consistent, since an op-

timal designed axiom-based probabilistic knowledge

base should have a conﬁdence value of σ(I ) ≈ 0.

5 EVALUATION

To test the quality of this approach, a crisp knowledge

base is used, which is called ”the solution” in this

section. The knowledge bases are generated by an

own software developed for this purpose, available

at https://github.com/unold/fcdc. Such a generated

knowledge base is called complete, i.e. there is

no further information inferable and it contains a

lot of redundant knowledge. From this complete

knowledge base, each axiom is changed to its nega-

tion with a probability of (1 − p) and p ∈ [0,1] is

assigned to each crisp axiom φ, so that it becomes a

probabilistic axiom p :: φ. The result is a damaged

knowledge base, where the probability of an axiom to

be false is p. By that, the knowledge base becomes a

probabilistic one, that exactly meets the requirements

to use the proposed approach.

For the requirements of this test, a simple random

optimization algorithm is used, which tries to ﬁnd a

good solution by randomly assuming some axioms as

true or false. It works in the following way.

maximumConfidenceApprox(KB) {

I = empty();

c = confidence(I);

repeat {

axiom = KB.getRandomAxiom();

I.add|I.remove(axiom);

if (I.consistent() & KB.confidence(I) > c)

c = KB.confidence(I);

else

I.remove|I.add(axiom);

} until c did not changed for all axioms

return c;

}

For very small knowledge bases, an exact algorithm

that determines the conﬁdence for every possible

world, is also used.

maximumConfidenceExact(KB) {

c = -1;

Axiom-based Probabilistic Description Logic

387

for (I subset KB) {

if (I.consistent() & KB.confidence(I) > c)

c = KB.confidence(I);

}

return c;

}

The aim of these algorithms is to ﬁnd an interpretation

I with the highest conﬁdence value for the damaged

knowledge base KB. The hope is, that this interpreta-

tion is also an interpretation for the correct solution.

However, in many cases there exists another interpre-

tation with a higher conﬁdence value.

The algorithms are executed on knowledge bases

with varying parameter p ∈ (0.5, 1). The parameter

p is only assigned to all assertional axioms of the

knowledge base and not to general concept inclusions,

since the assignment to general concept inclusions in-

volves many other axioms at the same time and there-

fore it is more likely to be considered false if its prob-

ability is less than 1. Also, the value p is only dis-

played in a range down to 50%, since a knowledge

base that contains only axioms that are more likely

to be false than to be true is rather useless, anyway.

The assumed correct solution, i.e. the solution be-

fore damaging the knowledge base, is always set to a

conﬁdence value of 0, since differences only appear

randomly.

Figure 1: The difference of conﬁdence (in percent) between

best and correct solution for different probabilities in small

knowledge bases of 100 (red) and 200 (blue) axioms.

Figure 1 depicts the difference between the cor-

rect solution and the one with the highest conﬁdence

value. It is determined by the exact algorithm, since

the knowledge bases consist of only 100 and 200 ax-

ioms, respectively. Therefore the best solution is al-

ways at least as good as the correct one. The horizon-

tal axis holds the value p for an axiom of the knowl-

edge base to be true. For instance, if p = 0.9 for a

knowledge base of 100 axioms, then in average 90 of

these 100 axioms are correct and 10 are incorrect.

A different result is depicted in ﬁgure 2, where

larger knowledge bases with 10000 and 20000 ax-

ioms are used, respectively. The exact algorithm is

Figure 2: The difference of conﬁdence (in percent) between

best and correct solution for different probabilities in large

knowledge bases of 10000 (red) and 20000 (blue) axioms.

not able to terminate within an appropriate time, thus

the heuristic algorithm has to be used. One can no-

tice, that it has more difﬁculties ﬁnding a good solu-

tion if there are many false axioms in the knowledge

base, which results in a conﬁdence value even worse

than the one of the correct solution, although there

might be even a solution with higher conﬁdence than

the correct one. Nevertheless, the results show that

the approach might work well for knowledge bases

with high trust, i.e. only a little amount of incorrect

information. In this case, it can detect the wrong in-

formation quite well.

For large knowledge bases with little trust, the

results show an increasing conﬁdence for the best

solution. Therefore it might be very likely that there

exist even better solutions in many cases, which are

simply not found by the algorithm. More sophisti-

cated heuristics should be developed to improve the

results. Nevertheless, unfortunately the correct world

is very likely to not be found, since there are other

worlds with higher conﬁdence.

Figure 3: The level of trust for the best solution for different

probabilities in knowledge bases of 100 (blue) and 10000

(red) axioms.

Another interesting aspect of this evaluation is

shown in ﬁgure 3. It depicts the amount of correct

information compared to the assumed correct solu-

tion for the solution with the highest conﬁdence value.

It clearly shows, that if a solution with higher conﬁ-

KEOD 2019 - 11th International Conference on Knowledge Engineering and Ontology Development

388

dence than the correct one is found, at least the trust of

this new knowledge base is higher than for the start-

ing point. Especially for an intermediate trust value

of around 60% to 75%, the improvement in trust is

noticeable. Probably the improvement is also better

for higher trust values p, if a better algorithm for the

optimization problem is used. An additional obser-

vation is, that the stability of values for large knowl-

edge bases is higher as compared to smaller knowl-

edge bases, i.e. the experimental results suffer from

high volatility for small datasets.

6 CONCLUSION

The paper presented a novel approach to deal with

probabilistic description logics. By not interpreting

each axiom separately, but a whole knowledge base

together, it enables results for cases, where other ap-

proaches would consider the knowledge base as in-

consistent. Also the computation time can be highly

reduced as compared to probabilistic description log-

ics by using optimization algorithms, which should be

subject to further research.

The disadvantage of the presented approach is,

that axioms can not be treated independently, i.e. the

whole knowledge base must be kept together. An-

other problem is that the probabilities of all axioms

are assumed to be independent of each other, i.e. there

is no space for redundancy. Nevertheless, the as-

sumption of having only one correct world instead

of assuming the existence of several possible worlds

seems more appropriate and realistic in many appli-

cation scenarios. Therefore a further development of

the presented idea seems useful.

For the evaluation, another interesting aspect

might be to consider also T-Box axioms to be uncer-

tain. Therefore another strategy for evaluation should

be developed, e.g. a different or separate treatment of

A-Box and T-Box axioms (Van Asch, 2013). For this

purpose, it might also be useful to change or param-

eterize formula (1) to determine the conﬁdence of an

interpretation for a given knowledge base in a differ-

ent way.

ACKNOWLEDGEMENTS

We would like to thank the Deutsche Forschungs-

gemeinschaft (DFG) and the agence nationale de la

recherche (ANR) for the support of the TEXTELSEM

project, in which this work has been done.

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